3 présentations

• 
  16h10 - 16h35

  Dynamic Systemic Cyber Risk Management under Incomplete Information

  • Juntao Chen, prés., New York University
  • Quanyan Zhu, New York University

  With the massive connections between different agents in the Internet network, cyber threats become ubiquitous and raise critical concerns for resource owners, e.g., data storage and cloud service providers. To address this issue, the owners can outsource their cyber risk management tasks to the professional security entities. In this paper, we use a principal-agent framework to capture the service relationships between two parties, i.e., the resource owner and the cyber risk manager. Specifically, we consider a dynamic systemic risk management problem with uncertainty where the owner only has the observations of cyber risk outcomes of the network rather than the efforts that the manager spends on protecting the resources. Under this incomplete information pattern, the owner aims to minimize the systemic cyber risks by designing a dynamic contract specifying the payment flows and the preferred efforts by taking the manager's incentives and rational behaviour into account. We obtain the optimal contracts by reformulating the problem into a stochastic optimal control program which can be solved using dynamic programming. We further investigate some special cases where the form of solutions can be characterized. Finally, some features of the optimal dynamic contracts are discussed including the information cost by comparing it with the one under complete information.

• 
  16h35 - 17h00

  On a Solution of a Guarantee Optimization Problem under a Functional Constraint on the Disturbance

  • Mikhail Gomoyunov, prés.,
  • Dmitriy Serkov, Krasovskii Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences

  A control problem for a dynamical system under disturbances is studied. A motion of the system is considered on a finite interval of time and is described by a nonlinear ordinary differential equation. The control is aimed at minimization of a given quality index. In addition to standard geometric constraints on the control and the disturbance, it is supposed that the disturbance satisfies a compact functional constraint. It means that all disturbance realizations that can happen in the system belong to some unknown set that is compact in the Lebesgue space. This notion of a functional constraint is quite general and can be used in order to formalize an available additional information about the properties of possible disturbance realizations as functions of time. Within the game-theoretical approach, the problem of optimizing the guaranteed result of the control is studied. For solving this problem, we propose a new construction of the optimal control strategy. This strategy can be considered as a control procedure with a guide. The proximity between motions of the original system and the guide is provided by a technique of dynamic reconstruction of the disturbance. The quality of the control process is attained due to the use of an optimal counter-strategy with full memory in the guide. The proposed control procedure makes it possible to develop numerical methods for solving the guarantee optimization problems under consideration. Examples are presented, results of numerical simulations are shown.

• 
  17h00 - 17h25

  On coincidence of value functions of time optimal games with and without life line

  • Nataly Munts, prés., Krasovskii Institute of Mathematics and Mechanics, UrB RAS
  • Sergey Kumkov, Institute of Mathematics and Mechanic, Russian Academy of Sciences

  This talk discusses time-optimal problems. Classic games, where the first player tries to guide the system to a prescribed closed target set as soon as possible, are considered along with games with lifeline, in which there is an additional closed set such that the second player wins if the system hits the set. Just the boundary of this set is called the lifeline. In the games of the last type, the cost functional can be formalized as the first instant, when the trajectory hits the target set, if the trajectory never leaves the game set, or plus infinity, if the trajectory never reaches the target set or comes to the lifeline. The existence of the value function of the classic time-optimal problem is proved, in particular, in works of N.N.Krasovskii and A.I.Subbotin. Earlier, the authors have proved the existence of the value function for the time-optimal game with lifeline based on results of these works. Furthermore, the authors have proved the existence of a generalized solution (minimax or viscosity) of the boundary value problem of the Hamilton-Jacoby equation corresponding to the game with lifeline and its coincidence with the value function of the game. Generally, this research was induced by a necessity to explore properties of the algorithm for constructing the value function of classic time-optimal games suggested by Italian mathematicians M.Bardi and M.Falcone. The theoretic proof of this method was carried out for the case when the entire unbounded game space is covered with a grid having an infinite number of nodes, whereas practical implementation can only deal with finite grids covering a bounded area, which corresponds to the game set in the case of the game with lifeline. The authors plan to present the results on coincidence of the value function of the games with and without lifeline. Also, it will be presented an estimate for the diameter of a ball, which must be located inside the set bounded by the lifeline to provide coincidence of the value functions at the prescribed point.